Analytical and numerical study for an integro-differential nonlinear Volterra equation

Guebbai, Hamza; Aissaoui, Mohamed Zine; Debbar, I.; Khalla, Besma

doi:10.1016/j.amc.2013.12.046

Cited by 13 publications

(11 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…As we can consider that this new Eq. (1) is the logic generalization of those studied by Guebbai et al (2014), sps3 (2020), sps4 (2020), and Touati et al (2019).…”

Section: Introductionmentioning

confidence: 82%

“…We add the following hypotheses for pure technical reasons : N (t, s, v) verify generalized Lipschitz condition. This kind of condition is very natural and has been used in several articles (sps1 2020; Guebbai et al 2014;sps3 2020;sps4 2020;Touati et al 2019;Linz 1985;Brunner 1989).…”

Section: Analytical Studymentioning

confidence: 99%

“…Our study starts with an analytical approach in which we are going to construct Lipschitz hypotheses, similar to those previously developed in sps1 (2020), Guebbai et al (2014), sps3 (2020), sps4 (2020), Touati et al (2019), Linz (1985), Brunner (1989), Babaaghaie and Maleknejad (2017), and Gu et al (2016) to ensure the existence and the uniqueness of the seismic function. These hypotheses are naturals and can be easily obtained in physics.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the weakly singular integro-differential nonlinear Volterra equation depending in acceleration term

et al. 2020

View full text Add to dashboard Cite

In this article, our study deals with the existence and the uniqueness of the solution of a second degree integro-differential nonlinear Volterra equation with a weakly singular kernel, i.e., the solution depends on its speed (first derivative) and its acceleration (second derivative); whereas using Nyström method and product integration method with piecewise projection, we approximate this solution. Keywords Volterra equation • Integro-differential • Fixed point • Nonlinear equation • Product integration method Mathematics Subject Classification 45D05 • 45G05 • 45J99 • 45E99 • 65R20 Recently sps1 (2020) published a physical model explaining seismic phenomenon in deterministic manner. They managed to construct numerical approximations with the reality. In their model, they proved that the seismic function which models the above-mentioned phenomena is the unique solution of the Volterra nonlinear integro-differential equation of the form: Communicated by Hui Liang.

show abstract

“…As we can consider that this new Eq. (1) is the logic generalization of those studied by Guebbai et al (2014), sps3 (2020), sps4 (2020), and Touati et al (2019).…”

Section: Introductionmentioning

confidence: 82%

Section: Analytical Studymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the weakly singular integro-differential nonlinear Volterra equation depending in acceleration term

et al. 2020

View full text Add to dashboard Cite

show abstract

“…However, this method was applied in the first state to approximate the solution of an integral equation and its theoretical framework is well given by Atkinson and Han in [13]. Recently, many others have relied on this method such as H. Guebbai et al in [14,15], H. Zhou and Q. Wang [16], S. Segni et al [17], M. Ghiat et al [18], S. Salah et al [19], Lemita and Guebbai [20]: to give an estimation solution of the non-linear integro-differential Volterra equation with a continuous kernel, or a weakly singular kernel [21,22] or to approximate a solution of a Fredholm linear system.…”

Section: Introductionmentioning

confidence: 99%

Solving linear Fredholm integro-differential equation by Nyström method

Tair¹,

Guebbai²,

Segni³

et al. 2021

J. Appl. Math. Comput. Mech.

View full text Add to dashboard Cite

The study of the solution's existence and uniqueness for the linear integrodifferential Fredholm equation and the application of the Nyström method to approximate the solution is what we will present in this paper. We use the Neumann theorem to construct a sufficient condition that ensures the solution's existence and uniqueness of our problem in the Banach space C 1 [a, b]. We have applied the Nyström method based on the trapezoidal rule to avoid adding other conditions in order to the approximation method's convergence. The Nyström method discretizes the integro-differential equation into solving a linear system. Only with the existence and uniqueness condition, we show the solution's existence and uniqueness of the linear system and the convergence of the numerical solution to the exact solution in infinite norm sense. We present two theorems to give a good estimate of the error. Also, to show the efficiency and accuracy of the Nyström method, some numerical examples will be provided at the end of this work.

show abstract

“…The identification accuracy of the non-linear system based on Volterra series in frequency domain is investigated in [10]. A non-linear analytic and numerical study of integral-differential equation for Volterra series is discussed in [11] while [12] presents the selection of general orthogonal bases for second-order Volterra filter.…”

Section: Introductionmentioning

confidence: 99%

Identification of a Non-Linear System Using Volterra Series Model with Calculated Kernels by Legendre Orthogonal Function

Mossadegh

Ghanbari

2017

IJAAS

View full text Add to dashboard Cite

Modeling and identification of non-linear systems have gained lots of attentions especially in industrial processes. Most of the actual systems have non-linear behavior and the first and simplest solution in modeling such systems is to linearize them which in most cases the result of linearization is not satisfactory. In this paper, modeling of non-linear systems is investigated using Volterra series model based on Legendre orthogonal function. Expansion of Volterra series kernels by Legendre orthogonal functions causes a reduction in the number of model parameters; hence, complexity of calculations would be decreased. Besides, if the free parameter is selected properly in these orthogonal functions, error is reduced and convergence speed of parameters is increased which leads to an increase in identification accuracy. In this paper, identification of non-linear system is presented with Volterra series expanded by Legendre function and PSO algorithm is used to calculate the optimum free parameters of Legendre function. Finally, in order to validate the efficacy and accuracy, the proposed algorithm is implemented on a non-linear system i.e. heat exchanger with actual data

show abstract

Analytical and numerical study for an integro-differential nonlinear Volterra equation

Cited by 13 publications

References 2 publications

On the weakly singular integro-differential nonlinear Volterra equation depending in acceleration term

On the weakly singular integro-differential nonlinear Volterra equation depending in acceleration term

Solving linear Fredholm integro-differential equation by Nyström method

Identification of a Non-Linear System Using Volterra Series Model with Calculated Kernels by Legendre Orthogonal Function

Contact Info

Product

Resources

About